Inductive Synthesis of Cover-Grammars with the Help of Ant Colony Optimization

A cover-grammar of a finite language is a context-free grammar that accepts all words in the language and possibly other words that are longer than any word in the language. In this paper, we describe an efficient algorithm aided by Ant Colony System that, for a given finite language, synthesizes (constructs) a small cover-grammar of the language. We also check its ability to solve a grammatical inference task through the series of experiments

Decomposing Finite Languages

The paper completely characterizes the primality of acyclic DFAs, where a DFA $\mathcal{A}$ is prime if there do not exist DFAs $\mathcal{A}_1,\dots,\mathcal{A}_t$ with $\mathcal{L}(\mathcal{A}) = \bigcap_{i=1}^{t} \mathcal{L}({\mathcal{A}_i})$ such that each $\mathcal{A}_i$ has strictly less states than the minimal DFA recognizing the same language as $\mathcal{A}$. A regular language is prime if its minimal DFA is prime. Thus, this result also characterizes the primality of finite languages. Further, the $\mathsf{NL}$-completeness of the corresponding decision problem $\mathsf{PrimeDFA}_{\text{fin}}$ is proven. The paper also characterizes the primality of acyclic DFAs under two different notions of compositionality, union and union-intersection compositionality. Additionally, the paper introduces the notion of S-primality, where a DFA $\mathcal{A}$ is S-prime if there do not exist DFAs $\mathcal{A}_1,\dots,\mathcal{A}_t$ with $\mathcal{L}(\mathcal{A}) = \bigcap_{i=1}^{t} \mathcal{L}(\mathcal{A}_i)$ such that each $\mathcal{A}_i$ has strictly less states than $\mathcal{A}$ itself. It is proven that the problem of deciding S-primality for a given DFA is $\mathsf{NL}$-hard. To do so, the $\mathsf{NL}$-completeness of $\mathsf{2MinimalDFA}$, the basic problem of deciding minimality for a DFA with at most two letters, is proven